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Unformatted text preview: (3/19/08) CHAPTER 2: THE DERIVATIVE AND APPLICATIONS The ancient Greeks did some amazing mathematics. Their work on the theory of proportions, in plane geometry, and on tangent lines, areas, and volumes came close to meeting modern standards of exposition and logic. Their explanations of topics from physics, such as the motion of projectiles and falling bodies, in contrast, were very different from modern theories. The philosopher Aristotle (384322 BC), for example, said that the motion of projectiles is caused by air pushing them from behind and that bodies fall with speeds proportional to their weights. (1) His point of view was generally not questioned in Western Europe until the Renaissance, when calculus was developed to study rates of change of nonlinear functions that arise in the study of motion and Newton explained how forces such as air resistance and gravity affect motion. We see in this chapter that the rate of change of a function is its derivative , which is the slope of a tangent line to its graph. We study linear functions and constant rates of change in Section 2.1 and average rates of change in Section 2.2. In Section 2.3 we give the general definition of the derivative as a limit of average rates of change. Formulas for exact derivatives of powers of x and of linear combinations of functions are derived in Section 2.4. In Section 2.5 we look at derivatives as functions and discuss applications that require finding approximate derivatives from graphs and tables. Rules for finding derivatives of products, quotients, and powers of functions are discussed in Sections 2.6 and 2.7. Section 2.8 deals with linear approximations and differentials. Section 2.1 Linear functions and constant rates of change Overview: As we will see in later sections, the rates of change of most functions are found by using calculus techniques to find their derivatives. In this section, however, we study applications that involve a special class of functions whose rates of change can be determined without calculus. These are the linear functions . A linear function is a first-degree polynomial y = mx = b . Its graph is a line and its (constant) rate of change is the slope of its graph. Topics: Constant velocity Other constant rates of change Approximating data with linear functions Constant velocity If an object moves on a straight path, we can use an s-axis along that path, as in Figure 1, to indicate the objects position. The objects s-coordinate is then a function of the time t , which is linear if the objects velocity is constant. s- 100 0 100 200 300 400 500 FIGURE 1 Example 1 A moving van is 200 miles east of a city at noon and is driving east at the constant velocity of 60 miles per hour. Give a formula for the vans distance s = s ( t ) east of the city t hours after noon....
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